Problem: $\overline{AB} = \sqrt{89}$ $\overline{AC} = {?}$ $A$ $C$ $B$ $\sqrt{89}$ $?$ $ \sin( \angle BAC ) = \frac{5\sqrt{89} }{89}, \cos( \angle BAC ) = \frac{8\sqrt{89} }{89}, \tan( \angle BAC ) = \dfrac{5}{8}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{AC}$ is adjacent to $\angle BAC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle BAC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{AC}}{\overline{AB}}= \frac{\overline{AC}}{\sqrt{89}} $ $ \overline{AC}=\sqrt{89} \cdot \cos( \angle BAC ) = \sqrt{89} \cdot \frac{8\sqrt{89} }{89} = 8$